Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
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A nice description of a specific toric variety
I am reading through Fulton's Introduction to Toric Varieties and working out some of the exercises. In chapter 1.4. we are asked to find the toric varieties associated to some specific fans on $N=\...
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Intersection in toric variety
In a toric variety $T$ of dimension $11$ I have a subvariety $W$ of which I would like to compute the dimension.
On $T$ there is a nef but not ample divisor $D$ whose space of sections has dimension $...
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Higher chow groups of affine toric varieties
Let $X$ be an affine toric variety defined over an algebraically closed field $k$ of characteristic zero.
I am trying to use Bloch’s Riemann-Roch Theorem for quasi-projective algebraic schemes in his ...
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Vanishing of chow group of 0-cycles for affine, simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero.
Let $X$ be an affine, simplicial toric variety over $k$.
If $X$ has dimension one, then it is the affine line over the field $k$, so ...
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A question related to the strong Oda conjecture
A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining ...
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Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form
Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
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Chow ring of simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a simplicial toric variety over $k$. In the 2011 book Toric Varieties by Cox, Little and Schenck, there is a theorem that ...
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Cohomology of equivariant toric vector bundles using Klyachko's filtration
I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties.
Whereas detailed literature ...
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Coordinate transformation for 3-dimensional simplicial cone in $\mathbb{R}^3$
Let $k$ be an algebraically closed field and let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$.Let $X$ be the affine toric variety over $k$ associated to the cone $\sigma$, i.e. set $X$...
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Equivariant line bundles over toric variety
Let $X$ be a projective $n$-dimensional toric variety acted by an algebraic torus $T\simeq \mathbb{C}^{\ast n}$. It is well known that any piecewise linear (and integer in some sense) function on ...
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Is this toric variety always smooth?
Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray in $\sigma$.
Let $U_{\rho}$ be defined as $\operatorname{Spec}(k[\...
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Is this closed subscheme a toric variety?
Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray of $\sigma$. Say $\rho=\sigma\cap H_m$, where $H_m$ is the plane in $...
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existence of moment maps for non-nef toric varieties
The noncompact toric variety $X_1 = \operatorname{Tot} \mathcal{O}(-1) + \mathcal{O}(-1) \to \mathbb{CP}^1$, the total space of the sum of two line bundles over the complex line, is defined as the ...
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Toric varieties as hypersurfaces of degree (1, ..., 1) in a product of projective spaces
I wonder if some hypersurfaces of multi-degree $(1, ..., 1)$ in a product of projective spaces are toric, with polytope a union of polytopes of toric divisors of the ambient space. This question is ...
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Three-dimensional analogues of Hirzebruch surfaces
There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
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Embedding toric varieties in other toric varieties as a real algebraic hypersurface
In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the ...
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Abstract definition of hypertoric varieties
I'm reading Proudfoot's survey on hypertoric varieties. In Section 1.4 he mentioned such a conjecture:
Conjecture 1.4.2 Any connected, symplectic, algebraic variety which is projective over its ...
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How to compute the higher G-theory of the weighted projective space $\mathbb{P}(1,1,m)$ using Mayer-Vietoris sequence?
Let $k$ be an algebraically closed field of characteristic zero.
Let $m$ be a positive integer and let $X$ be the weighted projective space $\mathbb{P}(1,1,m)$ over the field $k$.I am trying to ...
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Computing $G$-theory for a 3-dimensional affine simplicial toric variety
Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be the cone in $\mathbb{R}^3$ generated by $e_1,2e_1+e_2,e_1+2e_2+3e_3$.
Then it is easy to check that $\sigma$ is a 3-...
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K-theory of toric varieties
Let $X$ be a smooth projective toric variety over $\mathbb{C}$. Is there a good presentation for the K-theory ring $K_0(X)$ in terms of the corresponding fan, analogous to the presentation of the Chow ...
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Computing Grothendieck group of coherent sheaves of affine toric 3-fold from a simplicial cone
Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$. Let $X=\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^3])$ be the ...
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Log discrepancy of a toric exceptional divisor
Let $X$ be a toric singularity determined by a single cone. By taking a partition of the cone we may get a toric resolution of $X$.
Question: How can we compute the log discrepancies of the ...
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How can one determine the fan of a toric Weil divisor of a complete toric variety?
It's well known that there is a so-called cone-orbit correspondence between the cones in the fan of a toric variety and the orbits of the T-action on the toric variety. My question aims to understand ...
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Complex structures compatible with a symplectic toric manifold
Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action.
Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
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Reference Request: Classification of spherical varieties by "Weyl group invariant fans"
Apologies in advance for the vague question.
Let $X$ be a spherical variety with the action of some reductive group $G$. I have been told in conversation several times that such spherical varieties ...
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Finitely generated section ring of Mori dream spaces
Set-up: We work over $\mathbb{C}$. Let $X$ be a Mori dream space. Define, following Hu-Keel, the Cox ring of $X$ as the multisection ring
$$\text{Cox}(X)=\bigoplus_{(m_1\ldots,m_k)\in \mathbb{N}^k} \...
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References/applications/context for certain polytopes
First, let's consider an almost trivial notion. With any subspace $V\subset \mathbb R^n$ we associate a convex polytope $P(V)\subset V^*$ as follows. Each of the $n$ coordinates in $\mathbb R^n$ is a ...
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Zariski Cancellation and Toric Varieties, why isn't this affine variety toric?
The Zariski cancellation problem asks the following. If $ Y $ is a variety such that $ Y \times \mathbb{A}^{1}_{k} \cong \mathbb{A}^{n+1}_{k} $, then is $ Y $ isomorphic to $ \mathbb{A}^{n}_{k} $?
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Holomorphic cyclic action on smooth toric manifold extends to C^* action?
Let $Z_n$ be a homological trivial cyclic action on a smooth toric manifold compatible with the complex structure, the does it extends to a C^* action?
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Minimal model program for toroidal pairs
Suppose $(X, \Delta)$ be a toroidal pair over $Z$ where $f:(X, \Delta) \rightarrow (Z, \Delta_Z)$ is a toroidal morphism (see https://arxiv.org/pdf/alg-geom/9707012.pdf sections 1.2, 1.3 for the ...
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How to compute the $G$-theory of the weighted projective space $\mathbb{P}(1,1,2)$?
Let $k$ be an algebraically closed field of characteristic zero. Let $\Sigma$ be the fan in $\mathbb{R}^2$ consisting of three cones, cone generated by $e_1,e_2$,cone generated by $e_2,-e_1-2e_2$ and ...
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How to estimate the locus of non-zero cohomology for a equivariant toric reflexive sheaf, with a Klyachko description
I am trying to analyze Macaulay2 package "ToricVectorBundles". The package deals with equivariant reflexive sheaves on complete toric varieties. Such a sheaf is described by a set of ...
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Inverse image Weil divisor on a toric variety as a Cartier divisor
Let $X$ be a normal toric variety over an algebraically closed field and let $D$ be a torus invariant (prime) divisor. Assume $\pi\colon \tilde{X}\rightarrow X$ is a toric resolution of singularities ...
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Is a toric variety over a field of positive characteristic complete if and only if the support is all of $ N_{\mathbb{R}} $?
In Cox, Little and Schenck's book Toric Varieties they show that a toric variety $ X_{\Sigma} $ over a field of characteristic zero is complete if and only if the support is all of $ N_{\mathbb{R}} $. ...
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Kouchnirenko's theorem for non-generic polynomials
In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...
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Singularities of toric pairs
Suppose $(X,B)$ is a log canonical pair and $f: X \rightarrow Y$ an equidimensional toroidal contraction such that every component of $B$ is $f$ -horizontal. Let $\Gamma$ denote the reduced ...
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Strong factorisation conjecture for toric varieties
In this survey is remarked (see page 6 after Example 1.12) that to prove the
Conjecture 1.10 (Strong factorisation). Let $\phi: X \dashrightarrow Y $ be a birational map
between two quasi-projective ...
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Approximation of convex bodies by polytopes corresponding to smooth toric varieties
Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...
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Is there some kind of construction of a "canonical unirational variety" like the one for toric varieties?
Toric varieties in some sense a "canonical rational variety" in that one can construct them from purely combinatorial data and this combinatorial data makes it possible to turn many ...
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If $ Z $ is an $ n $-dimensional, projective variety, containing $ \mathbb{G}_{m}^{n} $, what is the obstruction to $ Z $ being toric?
Let $ Z $ be an $ n $-dimensional, projective, variety, over a field of arbitrary characteristic and let $ \iota: \mathbb G_{m}^{n} \to Z $ be a morphism such that for any $ z \in Z $, the fibre $ \...
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How to compute the $G$-theory of this simplicial toric surface?
Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma_0$ be the cone in $\mathbb{R}^2$ generated by $e_1,e_2$.And let $\sigma_1$ be the cone in $\mathbb{R}^2$ generated by $e_2,-...
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Cohomology of fibers of a morphism of a blowup of affine space
Consider $\mathbb A^n$ and let $\Sigma$ be a subdivision of its toric fan $\mathbb R^n_{\geq 0}$. This induces a toric blowup $\pi : Y \to \mathbb A^n$. Let $X \subseteq Y$ be the preimage of the ...
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Polytope of a projected toric variety
I was looking for such a result in the book by Cox, Little and Schenck but I'm not able to find a proper reference.
All of the following requirements are tacitly assumed to be in the projective ...
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Has anyone researched additive analogues of toric geometry in characteristic zero?
One definition of an $ n $-dimensional toric variety is that it is a variety $ Z $ for which there exists an equivariant embedding of
$ \mathbb{G}_{m}^{n} $ as a Zariski dense, open sub-variety of $ Z ...
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Facets of polytopes and toric morphisms
To every convex lattice polytope $P$ is associated a toric variety $X_P$, which can be realized as a projective variety.
Consider a facet $f$ of $P$, i.e. a codimension one boundary of the polytope.
...
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Why are symplectic toric varieties projective?
Let $X$ be a symplectic toric manifold meaning a compact symplectic manifold $(X, \omega)$ with $\dim{X} = 2n$ equipped with a Hamiltonian action of a maximal-dimension torus $\mathbb{T} = (\mathbb{S}^...
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What sort of spaces show up as intersection complexes of toric degenerations of Calabi-Yau Varieties?
Roughly, a toric degeneration is a proper flat family $f:\mathcal{X}\to D$ of relative dimension $n$ with the properties that $\mathcal{X}_t$ is an irreducible normal Calabi-Yau and $\mathcal{X}_0$ is ...
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Toric decomposition of multipartitions
Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$.
Let's call $\lambda$ ...
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Seeing $\mathbb{CP}^2 \mathbin\# \overline{\mathbb{CP}^2}$ as a symplectic reduction of different manifolds
I have been reading the paper "Remarks on Lagrangian intersections on toric manifolds" by Abreu and Macarini, which gives several non-displaceability results by avoiding the use of ...
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Does there exist a point $ x $ of an affine toric variety $ U_{\sigma} $ such that $ x $ is not compatibly split?
A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
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